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Register a new userFrom Quantum Mechanics to Quantum Field Theory: The Hopf route
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Added by
Gerard Henry Edmond Duchamp
Publication date
2010
fields
Physics
and research's language is
English
Authors
Allan I. Solomon
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We show that the combinatorial numbers known as {em Bell numbers} are generic in quantum physics. This is because they arise in the procedure known as {em Normal ordering} of bosons, a procedure which is involved in the evaluation of quantum functions such as the canonical partition function of quantum statistical physics, {it inter alia}. In fact, we shall show that an evaluation of the non-interacting partition function for a single boson system is identical to integrating the {em exponential generating function} of the Bell numbers, which is a device for encapsulating a combinatorial sequence in a single function. We then introduce a remarkable equality, the Dobinski relation, and use it to indicate why renormalisation is necessary in even the simplest of perturbation expansions for a partition function. Finally we introduce a global algebraic description of this simple model, giving a Hopf algebra, which provides a starting point for extensions to more complex physical systems.
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We consider the geometrization of quantum mechanics. We then focus on the pull-back of the Fubini-Study metric tensor field from the projective Hibert space to the orbits of the local unitary groups. An inner product on these tensor fields allows us to obtain functions which are invariant under the considered local unitary groups. This procedure paves the way to an algorithmic approach to the identification of entanglement monotone candidates. Finally, a link between the Fubini-Study metric and a quantum version of the Fisher information metric is discussed.
In this work we present a formalism of abstract quantum field theory for fat graphs and its realizations. This is a generalization of an earlier work for stable graphs. We define the abstract correlators $mathcal F_g^mu$, abstract free energy $mathcal F_g$, abstract partition function $mathcal Z$, and abstract $n$-point functions $mathcal W_{g,n}$ to be formal summations of fat graphs, and derive quadratic recursions using edge-contraction/vertex-splitting operators, including the abstract Virasoro constraints, an abstract cut-and-join type representation for $mathcal Z$, and a quadratic recursion for $mathcal W_{g,n}$ which resembles the Eynard-Orantin topological recursion. When considering the realization by the Hermitian one-matrix models, we obtain the Virasoro constraints, a cut-and-join representation for the partition function $Z_N^{text{Herm}}$ which proves that $Z_N^{text{Herm}}$ is a tau-function of KP hierarchy, a recursion for $n$-point functions which is known to be equivalent to the E-O recursion, and a Schrodinger type-equation which is equivalent to the quantum spectral curve. We conjecture that in general cases the realization of the quadratic recursion for $mathcal W_{g,n}$ is the E-O recursion, where the spectral curve and Bergmann kernel are constructed from realizations of $mathcal W_{0,1}$ and $mathcal W_{0,2}$ respectively using the framework of emergent geometry.
In this paper, we present a Hopf algebra description of a bosonic quantum model, using the elementary combinatorial elements of Bell and Stirling numbers. Our objective in doing this is as follows. Recent studies have revealed that perturbative quantum field theory (pQFT) displays an astonishing interplay between analysis (Riemann zeta functions), topology (Knot theory), combinatorial graph theory (Feynman diagrams) and algebra (Hopf structure). Since pQFT is an inherently complicated study, so far not exactly solvable and replete with divergences, the essential simplicity of the relationships between these areas can be somewhat obscured. The intention here is to display some of the above-mentioned structures in the context of a simple bosonic quantum theory, i.e. a quantum theory of non-commuting operators that do not depend on space-time. The combinatorial properties of these boson creation and annihilation operators, which is our chosen example, may be described by graphs, analogous to the Feynman diagrams of pQFT, which we show possess a Hopf algebra structure. Our approach is based on the quantum canonical partition function for a boson gas.
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Linking numbers appear in local quantum field theory in the presence of tensor fields, which are closed two-forms on Minkowski space. Given any pair of such fields, it is shown that the commutator of the corresponding intrinsic (gauge invariant) vector potentials, integrated about spacelike separated, spatial loops, are elements of the center of the algebra of all local fields. Moreover, these commutators are proportional to the linking numbers of the underlying loops. If the commutators are different from zero, the underlying two-forms are not exact (there do not exist local vector potentials for them). The theory then necessarily contains massless particles. A prominent example of this kind, due to J.E. Roberts, is given by the free electromagnetic field and its Hodge dual. Further examples with more complex mass spectrum are presented in this article.
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